A Ferrari statement after the Hungarian Grand Prix said that di Montezemolo "tweaked [Alonso's] ear" following a weekend in which the two-time Formula 1 champion had joked to Italian television reporters that he wanted a car as competitive as his rivals' for his birthday.
"I think it was wrong information coming to the president," Alonso said at Spa on Thursday.
"We looked back to all the press conferences - these days with recorders etc it's easy to check exactly what you say - and it was nothing different to any other grand prix and I tried to push for the best for the team and for our performance.
"He is the father of this family, of Ferrari, he is a great motivator. He told us exactly what he expects from us and what he wants from us."
It also emerged in Hungary that Red Bull was considering Alonso as a potential driver for 2014. Alonso confirmed that he intended to stay at Ferrari and denied that di Montezemolo questioned him about his intentions.
"I think the president is a very intelligent man to separate what is rumour and what can be true," Alonso said.
"We have both been in F1 for many years now and we know what the summer break means in terms of rumours so there were never questions."
Alonso believes the episode was a reminder of how the media focus on Ferrari can exacerbate the consequences of remarks that might go unnoticed in other teams.
"In anything we do, the repercussion of what Ferrari does is bigger than any other," he said.
"It was so clear in the young driver test. I said I was not going to the young driver test because there was nothing to test, there are no tyres to test and there is no setup possibility. [The press said] 'Alonso creates big polemics against Ferrari, against Pirelli, against the world!'
"But Kimi [Raikkonen] was not going and its [considered] funny. McLaren drivers were not going and it's funny.
"This is the repercussion of being Ferrari and the president and me talked about this as well.
"Also, we learned [about] doing four press conferences in the weekend in three different languages, two of them are not in my mother language and I am from the north, so don't speak perfect Spanish!
"It means there can be mistakes and there can be misunderstandings.
"What we need to do is to concentrate on racing because we need to be together, we need to be united and we need to win this championship.
"We need all the sides of the team in the right direction and also to pay maximum attention out of the car to not make any mistakes."
- Motor Racing